p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.410D4, C42.160C23, (C4×Q8).6C4, (C2×C4).58Q16, C4.35(C2×Q16), C4.96(C4○D8), C4○(C4.6Q16), C4.53(C2×SD16), C22⋊Q8.13C4, C4○2(C4.10D8), C4.10D8⋊44C2, C4⋊C8.256C22, C42.101(C2×C4), C4.6Q16⋊27C2, (C2×C4).102SD16, (C22×C4).233D4, C4⋊Q8.234C22, C4.34(Q8⋊C4), (C2×C42).204C22, C22.7(Q8⋊C4), C23.108(C22⋊C4), C42.12C4.22C2, C2.13(C23.24D4), C23.37C23.12C2, C2.15(M4(2).8C22), (C2×C4⋊C8).13C2, C4⋊C4.34(C2×C4), (C2×Q8).28(C2×C4), (C2×C4).1231(C2×D4), C2.13(C2×Q8⋊C4), (C22×C4).226(C2×C4), (C2×C4).154(C22×C4), (C2×C4).105(C22⋊C4), C22.218(C2×C22⋊C4), SmallGroup(128,274)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.410D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Subgroups: 188 in 106 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C4.10D8, C4.6Q16, C2×C4⋊C8, C42.12C4, C23.37C23, C42.410D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C4○D8, M4(2).8C22, C2×Q8⋊C4, C23.24D4, C42.410D4
►On 64 points
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 63 61 59)(58 64 62 60)
(1 14 58 24)(2 17 59 15)(3 16 60 18)(4 19 61 9)(5 10 62 20)(6 21 63 11)(7 12 64 22)(8 23 57 13)(25 40 56 41)(26 42 49 33)(27 34 50 43)(28 44 51 35)(29 36 52 45)(30 46 53 37)(31 38 54 47)(32 48 55 39)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 48 14 55 58 39 24 32)(2 50 17 43 59 27 15 34)(3 46 16 53 60 37 18 30)(4 56 19 41 61 25 9 40)(5 44 10 51 62 35 20 28)(6 54 21 47 63 31 11 38)(7 42 12 49 64 33 22 26)(8 52 23 45 57 29 13 36)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,63,61,59)(58,64,62,60), (1,14,58,24)(2,17,59,15)(3,16,60,18)(4,19,61,9)(5,10,62,20)(6,21,63,11)(7,12,64,22)(8,23,57,13)(25,40,56,41)(26,42,49,33)(27,34,50,43)(28,44,51,35)(29,36,52,45)(30,46,53,37)(31,38,54,47)(32,48,55,39), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,48,14,55,58,39,24,32)(2,50,17,43,59,27,15,34)(3,46,16,53,60,37,18,30)(4,56,19,41,61,25,9,40)(5,44,10,51,62,35,20,28)(6,54,21,47,63,31,11,38)(7,42,12,49,64,33,22,26)(8,52,23,45,57,29,13,36)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,63,61,59)(58,64,62,60), (1,14,58,24)(2,17,59,15)(3,16,60,18)(4,19,61,9)(5,10,62,20)(6,21,63,11)(7,12,64,22)(8,23,57,13)(25,40,56,41)(26,42,49,33)(27,34,50,43)(28,44,51,35)(29,36,52,45)(30,46,53,37)(31,38,54,47)(32,48,55,39), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,48,14,55,58,39,24,32)(2,50,17,43,59,27,15,34)(3,46,16,53,60,37,18,30)(4,56,19,41,61,25,9,40)(5,44,10,51,62,35,20,28)(6,54,21,47,63,31,11,38)(7,42,12,49,64,33,22,26)(8,52,23,45,57,29,13,36) );
G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,63,61,59),(58,64,62,60)], [(1,14,58,24),(2,17,59,15),(3,16,60,18),(4,19,61,9),(5,10,62,20),(6,21,63,11),(7,12,64,22),(8,23,57,13),(25,40,56,41),(26,42,49,33),(27,34,50,43),(28,44,51,35),(29,36,52,45),(30,46,53,37),(31,38,54,47),(32,48,55,39)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,48,14,55,58,39,24,32),(2,50,17,43,59,27,15,34),(3,46,16,53,60,37,18,30),(4,56,19,41,61,25,9,40),(5,44,10,51,62,35,20,28),(6,54,21,47,63,31,11,38),(7,42,12,49,64,33,22,26),(8,52,23,45,57,29,13,36)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C4○D8 | M4(2).8C22 |
| kernel | C42.410D4 | C4.10D8 | C4.6Q16 | C2×C4⋊C8 | C42.12C4 | C23.37C23 | C4×Q8 | C22⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.410D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 9 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 2 |
| 15 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,2,0,0,9,0,0,0,0,0,9,0,0,0,0,2],[15,0,0,0,0,9,0,0,0,0,0,9,0,0,15,0] >;
C42.410D4 in GAP, Magma, Sage, TeX
C_4^2._{410}D_4 % in TeX
G:=Group("C4^2.410D4"); // GroupNames label
G:=SmallGroup(128,274);
// by ID
G=gap.SmallGroup(128,274);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,184,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations